## SURFACE CRITICAL PHENOMENA

### Mykola Shpot

##### Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine

An extension of the massive field theory approach of Parisi to
systems with surfaces is presented. This approach provides the
opportunity to study the surface critical behavior directly in space
dimensions $d<4$ without having to resort to the $\epsilon$
expansion, especially in three dimensions. The method is elaborated
for the the semi-infinite $|{\phi}|^4$ $n$-vector model with a
boundary term $c_0\int_{\partial V}{\phi}^2$ in the action. To make
the theory UV finite in bulk dimensions $d<4$, a renormalization of
the surface enhancement $c_0$ is required, apart of the standard
mass renormalization; required norma\-lization conditions for the
renormalized theory are given. As a result, in addition to the the
usual bulk `mass' (the inverse correlation length) $m$, another mass
parameter appears in the theory, the renormalized surface
enhancement $c$. Thus the surface renormalization factors depend on
the renormalized coupling constant $u$ and the ratio $c/m$. The
special and ordinary surface transitions correspond to the limits
$m\to 0$ with $c/m\to 0$ and $c/m\to\infty$, respectively. The
surface critical exponents of the special and ordinary transitions
are given to one-loop order in $2\le d<4$ and to two-loop order at
$d=3$. The associated second order series expansions are analyzed by
Pad\'e-Borel summation techniques. The resulting numerical estimates
for the surface critical exponents are in good agreement with
available Monte Carlo simulations. This also holds for the surface
crossover exponent $\Phi$, for which the values $\Phi
(n\!=\!0)\simeq 0.52$ and $\Phi (n\!=\!1)\simeq 0.54$ are obtained,
considerably lower than the previous $\epsilon$-expansion estimates.

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